Is there a simple way to graph trig functions that aren't "regular"? Like f(t) = sin(t) is regular, but g(t) = 2sin(3(t-pi/3)) has different amplitude (2 instead of 1), a different period (pi instead of 2pi), and a phase shift (starting at pi/3 instead of at 0). It's complicated!!

anonmeans wrote:Is there a simple way to graph trig functions that aren't "regular"?

In finding the phase shift, the amplitude, and the period, you've actually already done the hard part!

Now draw the "regular" graph, but draw it "lightly", with the markings (like the t- and y-axes) in pencil. Draw lines for the amplitude (or tick-marks on the axis, if that's what your instructor prefers), and mark off the "regular" period "important" points; at the very least, mark off 0, pi/2, pi, 3pi/2, and 2pi. Do this for at least an extra half-period on either "end" of the "regular period".

Now re-label. Instead of the amplitude line being labelled as y = 1, re-label as y = 2. Instead of the period markings being 0, pi/2, etc, contract by labelling as 0, pi/4, etc (to account for the period being half as long as "regular"). Since this function is shifted pi/3 to the right, move all those labels by pi/3: 0 is relabelled as pi/3 (that is, you'll need to move the y-axis line back by pi/3 units), pi/4 is relabelled as pi/4 + pi/3 = 7pi/12, and so forth.

Now draw the shifted lines in darker, and erase the lines you don't need. You're done!